Question

How do you write the repeating decimal 0.5 as a fraction?

Answer 1

`0.55555.......=0.bar(5)=5/9`

Explanation:

There are two ways.

First consider `0.55555.......` as an infinite geometric series

`5/10+5/10^2+5/10^3+5/10^4+5/10^5+..................`

with first term `a+1=5/10` and common ratio `r=1/10`

Hence sum of the series is `a/(1-r)=(5/10)/(1-1/10)=(5/10)/(9/10)=5/9`

Second and easier is to consider `x=0.55555.......`

and hence `10x=5.55555..................`

and subtracting former from latter

`9x=5` i.e. `x=5/9`